Properties

Label 2-12e3-12.11-c1-0-11
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.03i·5-s − 0.267i·7-s − 3.86·11-s + 2.46·13-s + 6.69i·17-s + 1.73i·19-s + 5.93·23-s + 3.92·25-s − 2.07i·29-s + 0.535i·31-s − 0.277·35-s + 6.46·37-s + 2.07i·41-s − 7.46i·43-s + 9.52·47-s + ⋯
L(s)  = 1  − 0.462i·5-s − 0.101i·7-s − 1.16·11-s + 0.683·13-s + 1.62i·17-s + 0.397i·19-s + 1.23·23-s + 0.785·25-s − 0.384i·29-s + 0.0962i·31-s − 0.0468·35-s + 1.06·37-s + 0.323i·41-s − 1.13i·43-s + 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.664210206\)
\(L(\frac12)\) \(\approx\) \(1.664210206\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 1.03iT - 5T^{2} \)
7 \( 1 + 0.267iT - 7T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 5.93T + 23T^{2} \)
29 \( 1 + 2.07iT - 29T^{2} \)
31 \( 1 - 0.535iT - 31T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 + 7.45T + 59T^{2} \)
61 \( 1 - 9.39T + 61T^{2} \)
67 \( 1 - 9.73iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.153948574213536868641259244955, −8.506246931266579464788863271866, −7.901284392838942060188105930358, −6.96047712211576140302736192206, −5.97957743434529468043249569380, −5.30709215578723637930588097160, −4.33996389295935092616471326620, −3.42245376212811949010983934546, −2.24814175625428053578526816389, −0.952719425213923681364360251676, 0.860863024718210153044497222022, 2.61232254629017445791751252265, 3.04366525766328825292717262962, 4.46612007175428471745118419069, 5.21799325093062749756268774963, 6.08153331027514333131349925083, 7.13713524311385872332479935969, 7.52069426736546070043333932334, 8.647849207669574547154108268410, 9.240849013286114632052463319662

Graph of the $Z$-function along the critical line